Distinguish scalar and vector quantities and apply vector addition, subtraction and resolution into perpendicular components in one and two dimensions

What this dot point is asking

VCAA wants you to use the scalar/vector distinction with precision and to resolve and combine vectors using components.

Scalars and vectors

Vector addition

Graphical (head-to-tail). Place tail of second at head of first; resultant runs from tail of first to head of last.

Components. For a vector of magnitude $v$ at angle $\theta$ above the horizontal:

Sum $x$-components, sum $y$-components, recombine:

Vector subtraction

$\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. Reverse $\vec{b}$ and add. This is the key step for $\Delta \vec{v} = \vec{v}_f - \vec{v}_i$ in collisions and uniform circular motion.

Worked example

A car moves east at $20$ m s$^{-1}$ then turns to move north at $20$ m s$^{-1}$. Change in velocity:

$\Delta \vec{v} = 20 \hat\jmath - 20 \hat\imath$. Magnitude $\sqrt{20^2 + 20^2} = 28.3$ m s$^{-1}$.

Direction: $45°$ north of west.

The speed did not change but the velocity did. This is the source of centripetal acceleration in circular motion.

Common traps

Adding magnitudes of perpendicular vectors. $3$ m east plus $4$ m north is $5$ m displacement, not $7$ m.

Confusing speed and velocity. Speed is the magnitude of velocity. A car turning at constant speed has changing velocity.

Mixing degrees and radians on the calculator. VCE Physics expects degrees unless specified.

In one sentence

Scalars have magnitude only; vectors have magnitude and direction (displacement, velocity, force, momentum) and are combined by head-to-tail addition or by resolving into perpendicular components, with subtraction performed by reversing the subtracted vector.